Estimates on conjectures of Minkowski and woods

Abstract

Let Rⁿ be the n-dimensional Euclidean space. Let Λ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of Λ other than the origin O and has n linearly independent points of Λ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in Rⁿ of radius √n/4 contains a point of Λ. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il'in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms.